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ON THE RELATIONSHIP BETWEEN MINIMAL LATTICE KNOTS AND MINIMAL CUBE KNOTS

CASEY MANN

The University of Texas at Tyler, 3900 University Blvd, Department of Mathematics, HPR 102, Tyler, TX 75799, USA

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and

JENNIFER MCLOUD-MANN

The University of Texas at Tyler, 3900 University Blvd, Department of Mathematics, HPR 102, Tyler, TX 75799, USA

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https://doi.org/10.1142/S0218216505004111Cited by:1 (Source: Crossref)

Abstract

Lattice knots have been studied in recent years, especially in ℤ³ and with respect to how many edges are required to form a knot. Knots formed from cubes have also been investigated for their ability to tessellate space. In this article, we demonstrate that there is a relationship between the minimum number of edges required to form a lattice knot and the minimum number of cubes required to form the same kind of knot. We further investigate the relationship in the face-centered cubic lattice.

Keywords:

AMSC: 57M27, 57C07

References

Y. Diao, J. Knot Theory Ramifications 2, 413 (1993). Link, Web of Science, Google Scholar
E. J. J. van Rensburg, J. Knot Theory Ramifications 4, 115 (1995). Link, Web of Science, Google Scholar
E. J. J. van Rensburg and S. G. Whittington, J. Phys. A: Math. Gen. 23, 3573 (1990). Google Scholar
E. J. J. van Rensburg, Ideal Knots, Series on Knots and Everything 19 (World Scientific Publishing, River Edge, NJ, 1998) pp. 88–106. Link, Google Scholar
C. Adams, Math. Intelligencer 17(2), 41 (1995). Web of Science, Google Scholar
S. Oh, J. Knot Theory Ramifications 5(3), 405 (1996). Link, Web of Science, Google Scholar
M. Senechal , Quasicrystals and Geometry ( Cambridge University Press , New York , 1995 ) . Google Scholar
E. S. Fedorov, Nachala Ucheniya o Figurakh, Notices of the Imperial St. Petersburg Mineralogical Society, 2nd Series, Vol. 21, pp. 1–279 . Google Scholar